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A010790 a(n) = n!*(n+1)!. 33
1, 2, 12, 144, 2880, 86400, 3628800, 203212800, 14631321600, 1316818944000, 144850083840000, 19120211066880000, 2982752926433280000, 542861032610856960000, 114000816848279961600000, 27360196043587190784000000, 7441973323855715893248000000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Let M_n be the symmetrical n X n matrix M_n(i,j)=1/min(i,j); then for n>=0 det(M_n)=(-1)^(n-1)/a(n-1). - Benoit Cloitre, Apr 27 2002

If n women and n men are to be seated around a circular table, with no two of the same sex seated next to each other, the number of possible arrangements is a(n-1). - Ross La Haye, Jan 06 2009

a(n-1) is also the number of (directed) Hamiltonian cycles in the complete bipartite graph K_{n,n}. - Eric W. Weisstein, Jul 15 2011

a(n) is also number of ways to place k nonattacking semi-bishops on an n X n board, sum over all k>=0 (for definition see A187235). - Vaclav Kotesovec, Dec 06 2011

a(n) is number of permutations of {1,2,3,...,2n} such that no odd numbers are adjacent. - Ran Pan, May 23 2015

a(n) is number of permutations of {1,2,3,...,2n+1} such that no odd numbers are adjacent. - Ran Pan, May 23 2015

REFERENCES

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, pp. 63-65.

Kenneth H. Rosen, Editor-in-Chief, Handbook of Discrete and Combinatorial Mathematics, CRC Press, 2000, page 91. [Ross La Haye, Jan 06 2009]

LINKS

T. D. Noe, Table of n, a(n) for n = 0..100

J. Agapito, On symmetric polynomials with only real zeros and nonnegative gamma-vectors, Linear Algebra and its Applications, Volume 451, 15 June 2014, Pages 260-289.

V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 268.

S. Tanimoto, Parity alternating permutations and signed Eulerian numbers, Ann. Comb. 14 (2010) 355 (total number of PAPs of [2n+1].)

Eric Weisstein's World of Mathematics, Complete Bipartite Graph

Eric Weisstein's World of Mathematics, Hamiltonian Cycle

Index entries for sequences related to factorial numbers

FORMULA

Integral representation as n-th moment of a positive function on a positive half axis, in Maple notation: a(n)=int(x^n*2*sqrt(x)*BesselK(1, 2*sqrt(x)), x=0..infinity), n=0, 1... Hypergeometric g.f.: a(0)=1, a(n)=subs(x=0, n!*diff(1/((x-1)^2), x$n)), n=1, 2... - Karol A. Penson, Oct 23 2001

Sum_{i >=0} 1/a(i) = A096789. - Gerald McGarvey, Jun 10 2004

With b(n)=A002378(n) for n>0 and b(0)=1, a(n) = b(n)*b(n-1)...*b(0). - Tom Copeland, Sep 21 2011

a(n) = det(PS(i+1,j), 1 <= i,j <= n), where PS(n,k) are Legendre-Stirling numbers of the second kind. - Mircea Merca, Apr 04 2013

a(n) = (2*n)! / A000108(n) which implies that the e.g.f. of A126120 is Sum_{k>=0} x^(2*k) / a(k). - Michael Somos, Nov 15 2014

0 = a(n)*(+18*a(n+2) - 15*a(n+3) + a(n+4)) + a(n+1)*(-9*a(n+2) - 4*a(n+3)) + a(n+2)*(+3*a(n+2)) for all n>=0. - Michael Somos, Nov 15 2014

From Ilya Gutkovskiy, Jan 20 2017: (Start)

a(n) ~ 2*Pi*n^(2*n+2)/exp(2*n).

Sum_{n>=0} (-1)^n/a(n) = BesselJ(1,2) = 0.576724807756873387202448... (End)

EXAMPLE

G.f. = 1 + 2*x + 12*x^2 + 144*x^3 + 2880*x^4 + 86400*x^5 + ...

MAPLE

f:= n-> n!*(n+1)!: seq(f(n), n=0..30);

MATHEMATICA

s=1; lst={s}; Do[s+=(s*=n)*n; AppendTo[lst, s], {n, 1, 4!, 1}]; lst (* Vladimir Joseph Stephan Orlovsky, Nov 15 2008 *)

Times@@@Partition[Range[0, 25]!, 2, 1] (* Harvey P. Dale, Jun 17 2011 *)

PROG

(Sage) [stirling_number1(n, 1)*factorial (n-2) for n in xrange(2, 17)] # Zerinvary Lajos, Jul 07 2009

(PARI) a(n)= n!^2*(n+1) \\ Charles R Greathouse IV, Jul 31 2011

(MAGMA) [Factorial(n)*Factorial(n+1): n in [0..20]]; // Vincenzo Librandi, Aug 08 2014

CROSSREFS

Cf. A004737, A000290.

Second column of triangle A129065.

Cf. A000108, A126120.

Sequence in context: A262241 A052742 A035049 * A221101 A187748 A086928

Adjacent sequences:  A010787 A010788 A010789 * A010791 A010792 A010793

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified September 26 06:24 EDT 2017. Contains 292502 sequences.